magic tour

Suppose a chess piece makes a tour on an n
× n chessboard whose squares are numbered from 1 to n2
along the path of the piece. The tour is a magic tour if the resulting arrangement
of numbers is a magic square, and is a semimagic
tour if the resulting arrangement of numbers is a semi-magic
square. Magic knight's tours aren't
possible on n × n boards if n is odd. They are
possible for all boards of size 4k × 4k for k
> 2, but are believed to be impossible for n = 8.

Magic tours have been found in 4 × 4 × 4, 8 × 8 × 8,
and 12 × 12 × 12 cubes, and on the surface of 8 × 8 ×
8 cube. However, there are no known knight tours in hypercubes.

[Thanks to Awani Kumar for corrections to and additional
information for this entry.]